I Is acceleration absolute or relative - revisited

[PeterDonis]

I can't shake that proper acceleration (and the resulting observation) is invarient and MUST be present in all reference frames. If the proper rotating universe is a legitimate reference frame

You're mixing up the concept of "reference frame" with the concept of "spacetime geometry".

General covariance means that all reference frames are equivalent for describing the same spacetime geometry. So if a given object in a given spacetime geometry has a given proper acceleration in one frame, it will have the same proper acceleration in all frames.

However, when you talk about a proper rotating universe, as compared to a proper non-rotating universe, you are talking about two different spacetime geometries. It doesn't even make sense to compare reference frames in these two different spacetime geometries (except locally). Reference frames that describe things in a proper rotating universe have nothing whatever to do with reference frames that describe things in a proper non-rotating universe.

 

[Dale]

Are you referring to the gravitational constant $G$?

No, I was referring to the cosmological constant.

I guess you could talk about tuning G, but that is basically an artifact of the units, or at least it is fixed so that we get Newtonian gravity as a limit of GR. So I would tend not to list G as a tuneable parameter of GR.

 

[Dale]

Is a rotating universe a legitimate frame of reference ? If the answer is yes, then given proper acceleration (and resulting observation) is invarient, then it (plus the corresponding observation) must by definition also exist in the rotating universe reference frame. Therefore your answer can only be no, a rotating universe is not a legitimate reference frame ?

Yes, it is legitimate. In technical language, you can make a tetrad that is at every event comoving with the local matter.

Yes, the proper rotation of the universe exists in every frame, meaning that an accelerometer at rest with respect to the universe will measure rotation. Regardless of whether you are describing things in the frame described above or in some other frame.

I am unclear why you think my answer must be “no”.

 

[Peter Leeves]

You're mixing up the concept of "reference frame" with the concept of "spacetime geometry".

I've learned in this thread that proper (true ?) acceleration is invarient and thus applies in all reference frames, including inertial ones. It's detected by an accelerometer, and all physics and corresponding results would be the same to all observers no matter what their reference frame. Was that wrong ? If it's correct, then I have at least a rudimentary grasp on "reference frame".

Spacetime geometry I understand as a fabric comprising one dimension of time and three dimensions of space. Within this fabric, mass determines the geometry of the three spatial dimensions, and the geometry of the three spatial dimentions determines how mass moves within it. Properties of mass, energy and light within spacetime are determined by Special Relativity (constant velocity) and General Relativity (acceleration, or via equivalence, gravity). Is that wrong or incomplete ? If so, I'm always happy to learn.

General covariance means that all reference frames are equivalent for describing the same spacetime geometry. So if a given object in a given spacetime geometry has a given proper acceleration in one frame, it will have the same proper acceleration in all frames.

Took several re-reads, but I now understand General Covariance. Thanks.

I may understand better now. Can you please confirm you mean "stationary universe" is one spacetime geometry, and all reference frames WITHIN THAT GEOMETRY must show the same proper acceleration (as measured by an accelerometer). But if you jumped into a DIFFERENT spacetime geometry "rotating universe" it wouldn't show the same (original) proper acceleration ?

However, when you talk about a proper rotating universe, as compared to a proper non-rotating universe, you are talking about two different spacetime geometries. It doesn't even make sense to compare reference frames in these two different spacetime geometries (except locally). Reference frames that describe things in a proper rotating universe have nothing whatever to do with reference frames that describe things in a proper non-rotating universe.

If my understanding of different spacetime geometries is now correct, then yes, I can now follow each and every point in this paragraph.

However, I'm uncertain as to whether "rotating universe" is a different spacetime to "stationary universe". This may be blindingly obvious to you. Sorry, I need to think about it. My immediate thought is it's actually the same spacetime geometry, simply rotating (as a thought experiment) and hence invariant acceleration would still be applicable and result in indentical observation.

Could you please explain why it's a different spacetime to the original one ? I have an argument for it being the original one - but rather think about it a bit.

 

[PeterDonis]

I was referring to the cosmological constant.

I would view this as a solution-specific parameter, and if we're considering those, we have to consider the entire effective stress-energy tensor (which the cosmological constant would be part of--it's just a piece of the SET of the form $\Lambda g_{\mu \nu}$), which gives a total of ten tuneable parameters.

 

[Dale]

I would view this as a solution-specific parameter, and if we're considering those, we have to consider the entire effective stress-energy tensor (which the cosmological constant would be part of--it's just a piece of the SET of the form $\Lambda g_{\mu \nu}$), which gives a total of ten tuneable parameters.

I would not consider the sources to be tuneable parameters of the theory. I can see your point that the cosmological constant could be considered part of the sources rather than a parameter of the theory. I know many people do that so you are at least in good company, but I have always felt that it deserves its own spot outside of the SET. I don't have a particularly good reason for that, other than the way that I was first exposed to GR.

In any case, since I would not consider the sources to be parameters of the theory then I would say that if you want to include the cosmological constant in the sources then GR would have no tuneable parameters at all.

 

[PeterDonis]

I've learned in this thread that proper (true ?) acceleration is invarient and thus applies in all reference frames, including inertial ones. It's detected by an accelerometer, and all physics and corresponding results would be the same to all observers no matter what their reference frame. Was that wrong ?

No. But "all reference frames" here really means "all reference frames that describe the same spacetime geometry".

Spacetime geometry I understand as a fabric comprising one dimension of time and three dimensions of space.

More precisely, one of an infinite number of possible such "fabrics", described by different physically distinct solutions of the Einstein Field Equation. The flat Minkowski spacetime of special relativity, which is what we have implicitly been using the term "non-rotating universe" to mean, is one such solution. The curved spacetime of the Godel universe, which is what we have implicitly been using the term "rotating universe" to mean (once you clarified that that's what you intended, in a post I'll quote below), is another, different such solution. So they are two different spacetime geometries.

Within this fabric, mass determines the geometry of the three spatial dimensions

No. Mass (more precisely, stress-energy) determines the geometry of all four dimensions. The three spatial dimensions by themselves do not have a well-defined geometry. Only the spacetime, consisting of all four dimensions, does.

Properties of mass, energy and light within spacetime are determined by Special Relativity

Only if the spacetime is flat Minkowski spacetime. See below.

(constant velocity) and General Relativity (acceleration, or via equivalence, gravity).

No. You have to use GR whenever spacetime is curved. But you can have acceleration (proper acceleration) in flat spacetime, and SR can handle that just fine. And you can have geodesic motion (free fall, zero proper acceleration) in curved spacetime, and you need GR to handle that.

Can you please confirm you mean "stationary universe" is one spacetime geometry, and all reference frames WITHIN THAT GEOMETRY must show the same proper acceleration (as measured by an accelerometer).

Yes, this is correct. More precisely, once you've specified a particular state of motion for an object in a given spacetime geometry, that fixes its proper acceleration, which will be the same in all reference frames that describe that spacetime geometry.

if you jumped into a DIFFERENT spacetime geometry "rotating universe" it wouldn't show the same (original) proper acceleration ?

Correct. More precisely, you would need to specify a particular state of motion of an object in this different spacetime geometry, and once you did that, that would fix the proper acceleration of the object, which would be the same in all reference frames that describe this different spacetime geometry. But you would have no reason in general to expect that this proper acceleration would be the same as the one in the previous case above. And even if it was, that wouldn't necessarily tell you anything useful.

I'm uncertain as to whether "rotating universe" is a different spacetime to "stationary universe".

I thought the previous discussion had made that clear. It is.

More precisely, when you answered that your intent was (2) here...

I meant (2), not just coordinate change.

...you were answering the question quoted above, giving the answer "yes".

 

[PeterDonis]

I have always felt that it deserves its own spot outside of the SET.

Historically, it did have such a separate spot, yes. However, the fact that the value of the cosmological constant in our actual universe has to be determined from observation, just like the rest of the stress-energy distribution, to me means it should be considered to be part of the sources, i.e., as a solution-dependent parameter just like the rest of the SET.

If there were a way of theoretically deriving a value for the CC independently of the rest of the SET, or, to put it another way, independently of any properties of matter/energy, that would be different; but AFAIK there isn't one. The only theoretical method we have for deriving a value for the CC is from the QFT vacuum, and that method is not independent of properties of matter/energy, since it depends on which specific quantum fields are present, and that depends on what kinds of matter/energy we include in our theory. An "independent" method of deriving a value of the CC, as I am using the term, would do it purely from properties of spacetime geometry, without having to make use of any properties of other fields.

 

[Peter Leeves]

I am unclear why you think my answer must be “no”.

For a moment, put aside the question of whether we are talking about 1 spacetime covering both scenarios (rotating bucket and rotating universe) or 2 spacetimes (different spacetime required for each scenario). Up to this point I didn't realize the implications - but now I'm starting to.

So, assuming both scenarios are covered by the same spacetime geometry, we know proper acceleration (and corresponding observation) is invariant and MUST apply in all reference frames (including inertial ones). If a stationary universe and rotating universe are both legitimate reference frames, then any proper acceleration (and corresponding observation) in one MUST also be identical in the other. Ergo, if an accelerometer in stationary universe registers a proper acceleration (and the water bulges), it necessarily follows that an identical proper acceleration (and water bulge) MUST also occur in the rotating universe - because the acceleration (and observation) is invariant.

By agreeing the rotating universe is a legitimate reference frame, you have agreed that the observation (water bulge) will occur in both stationary and rotating universes (because proper acceleration is invariant and MUST occur in both). As you've been arguing up till now that the same observation wouldn't be seen in the rotating universe, of course I expected you to say it's because the rotating reference frame isn't legitimate.

 

[Dale]

However, the fact that the value of the cosmological constant in our actual universe has to be determined from observation, just like the rest of the stress-energy distribution, to me means it should be considered to be part of the sources, i.e., as a solution-dependent parameter just like the rest of the SET.

I can see your point, but on the other hand, in principle once you have determined the cosmological constant from cosmological observations then in principle a star's gravitational field should be described with a spherical distribution of matter using equations including the cosmological constant. I.e. you would take the cosmological constant (once determined) as part of the equations of the theory that you would use with any distribution of matter.

 

[Dale]

So, assuming both scenarios are covered by the same spacetime geometry,

I cannot do that. It is a false assumption and anything can be proven from a false assumption.

They are two different scenarios with different spacetime geometries represented by different metrics. This is not something that can be hypothesized away because it is central to the whole topic. The case where the universe is rotating around a non-rotating observer is physically distinct from the case where an observer is rotating within a non-rotating universe (all rotations "proper" as described above). There is no coordinate transform between the two cases because it is a completely different spacetime. Within each spacetime all frames are equally valid, but the two spacetimes do not share frames or overlap in any way, they are wholly distinct.

 

[PeterDonis]

you would take the cosmological constant (once determined) as part of the equations of the theory that you would use with any distribution of matter.

No, actually that's not what we do. We don't model the solar system (or binary pulsars, or galaxies, or even galaxy clusters) using a CC in the equations. It's true that that is because the value of the CC we determine from cosmological observations (the acceleration of the universe's expansion) is much too small to have any detectable effect on solar system observations (or even on observations of other systems like the ones I listed); but that's just another way of saying that we use observations to determine what CC parameter we use in our models, just as with the rest of the SET. If we had observations accurate enough to detect CC effects on smaller systems such as galaxies or solar systems, we would be able to test by observation whether the CC is actually a constant everywhere or whether it varies, and in our solar system models we would be using whatever CC parameter was shown by those observations, even if it was different from the one we use for our models of the universe as a whole based on cosmological observations.

 

[Peter Leeves]

I cannot do that. It is a false assumptions and anything can be proven from a false assumption.

They are two different scenarios with different spacetime geometries represented by different metrics. This is not something that can be hypothesized away because it is central to the whole topic.

Understood. That's why I stressed "assuming" - because I've just begun to understand the relevance of the two scenarios being in separate spacetimes. If it's two spacetime geometries, I now understand why you won't get the same observation in the rotating universe.

I have an argument I'd like to put that the two scenarios occur in the same spacetime geometry. I need to think on it a little and will post soon.

 

[PeterDonis]

you've been arguing up till now that the same observation wouldn't be seen in the rotating universe, of course I expected you to say it's because the rotating reference frame isn't legitimate.

No, it's because there are two different spacetime geometries.

put aside the question of whether we are talking about 1 spacetime covering both scenarios (rotating bucket and rotating universe) or 2 spacetimes (different spacetime required for each scenario).

We can't. That question must be answered before we can answer anything else.

Up to this point I didn't realize the implications - but now I'm starting to.

Yes, and that means you should take a big step back and rethink things. See below.

I have an argument I'd like to put that the two scenarios occur in the same spacetime geometry.

You can, of course, define the phrase "non-rotating bucket in a rotating universe" to mean "the same spacetime geometry as the rotating bucket in a non-rotating universe, just described in different coordinates". And if you do that, then the obvious answer to your question will be that the observed shape of the water in the bucket will be the same in both cases. But that answer has nothing to do with physics; it just has to do with how you defined the phrase "non-rotating bucket in a rotating universe".

 

[Dale]

I have an argument I'd like to put that the two scenarios occur in the same spacetime geometry. I need to think on it a little and will post soon.

If you put the two scenarios in the same spacetime geometry then the rotation will be coordinate rotation rather than proper rotation.

 

[Peter Leeves]

No. But "all reference frames" here really means "all reference frames that describe the same spacetime geometry".

Got it. One spacetime geometry can contain multiple reference frames and all will agree on those physics and observations. Another spacetime geometry can also contain multiple reference frames and all will agree on those physics and observations. But a reference frame in one spacetime has no (or very little, depending on the relationship between the two I guess) relevance to the reference frame in the other.

When I answered (2) earlier, it's now apparent I didn't fully understand the implications. Hopefully I have a better grasp now.

 

[Peter Leeves]

You can, of course, define the phrase "non-rotating bucket in a rotating universe" to mean "the same spacetime geometry as the rotating bucket in a non-rotating universe, just described in different coordinates". And if you do that, then the obvious answer to your question will be that the observed shape of the water in the bucket will be the same in both cases. But that answer has nothing to do with physics; it just has to do with how you defined the phrase "non-rotating bucket in a rotating universe".

I need to re-read this one several times. It's very interesting.

My immediate thought is it's similar (not identical) to what Einstein did in his linear thought experiement. I agree it's nothing to do with physics, yet it still provided a valid explanation of observations. This is all I've been trying to do since the OP. Someone was arguing if the water is stationary, then there's no reason for the it to climb the walls of the bucket. I was trying to provide the missing explanation why it would. I do think you've described the solution. Static water is just a different coorinate system in the same spacetime geometry. The acceleration is invariant in both reference frames and hence the observation has to be identical.

Einstein's thought experiment may not have been physics but it nevertheless added two things of value. Firstly he described a mechanism by which one scenario was equivalent to the other. Secondly he concluded that both viewpoints were equally valid. This answer does the same (though definitely NOT using the same or even similar mechanism).

And I've learned an awful lot in arriving at the correct answer. Thank you.

I have a sneaking suspicion you knew this all along and watched me slog my way through it - ha ha ha. Well played gents, well played

 

[PeterDonis]

One spacetime geometry can contain be described by multiple reference frames

See correction above. Reference frames aren't "contained" in spacetime geometries. They are abstract things constructed by humans to describe spacetime geometries.

 

[Peter Leeves]

Now I understand a little better, I wish to make a correction (must be hundreds required, lol). Too many times in this thread I referred to stationary or rotating "bucket", or stationary or rotating "bucket/water". I think I'm correct to say that only the water is stationary (defining it's reference frame). The bucket must rotate along with the rest of the universe.

 

[PeterDonis]

My immediate thought is it's similar (not identical) to what Einstein did in his linear thought experiement.

This was another case of defining different reference frames to describe the same spacetime geometry, yes. The spacetime geometry is flat spacetime; one reference frame is the usual global inertial frame; the other reference frame is the "accelerated" frame in which the linearly accelerating "elevator" is at rest (today we would call this frame "Rindler coordinates", which you can look up for more info). In the latter frame, a "gravitational field" exists, in the sense @Dale defined for that term--there are nonzero Christoffel symbols--and in the sense that a freely falling object will, relative to the frame, have a "downwards" coordinate acceleration, while an object at rest in the frame will have an "upwards" proper acceleration.

The part that might have confused you is that Einstein then went on to draw an analogy with a similar local situation in a different spacetime geometry, the curved spacetime geometry around a planet like the Earth. Sitting at rest inside a room on the surface of the planet "looks" locally like sitting at rest inside the linearly accelerating "elevator" in flat spacetime described above. But what "locally" means here is that we are ignoring the curvature of the spacetime geometry and restricting attention to a small patch of it, small enough that we can treat it as flat. Which means that we are really treating it, for this restricted local purpose, as the same spacetime geometry as the flat spacetime case above. So this is not actually a case of "the same" scenario in two different spacetime geometries. It's just a case of two different choices of coordinates/reference frames in the same (local) spacetime geometry.

 

[PeterDonis]

I have a sneaking suspicion you knew this all along and watched me slog my way through it

Not at all. We have been trying to figure out (or help you to figure out) what you actually mean. You appear to have changed your mind about that twice now.

 

[PeterDonis]

I think I'm correct to say that only the water is stationary (defining it's reference frame). The bucket must rotate along with the rest of the universe.

No. The bucket and the water are at rest relative to each other in every scenario we have discussed. (Note that we have been ignoring any process of "spinning up" the bucket, i.e., we have been ignoring transients and only considering steady-state situations of one sort or another. During transients the bucket and water can of course move relative to each other; but all such relative motion must disappear by the time a steady state has been reached.)

 

[Peter Leeves]

Not at all. We have been trying to figure out (or help you to figure out) what you actually mean. You appear to have changed your mind about that twice now.

I'm happy to accept that.

Ultimately I wanted to answer "Is acceleration absolute or relative". The answer is, absolute (invariant) no matter which reference frame is used or whether they are inertial or non-inertial, but only within a single spacetime geometry.

Specifically, I wanted to answer, if the water is stationary what would cause the water to climb the walls of the bucket. The answer is, the stationary water is just a different coorinate system (reference frame) in the same spacetime geometry. The acceleration is invariant in all reference frames and hence the observation has to be identical.

I bet you can't wait for my post on the delayed choice quantum eraser experiment

 

[Peter Leeves]

No. The bucket and the water are at rest relative to each other in every scenario we have discussed.

Disagree. The bucket and the water are NOT at rest relative to each other in every scenario we've discussed. The water has to spin up in every scenario and therefore the reference frame can only be with respect to the non-rotating water. If I found my opinion changed by the end of the thread, then potentially anyone else reading this thread could miss the same distinction. It's worth including for accuracy and completeness (things we all value) and costs nothing. I say this even given the immediately following proviso, which is excellent and provides the description required.

(Note that we have been ignoring any process of "spinning up" the bucket, i.e., we have been ignoring transients and only considering steady-state situations of one sort or another. During transients the bucket and water can of course move relative to each other; but all such relative motion must disappear by the time a steady state has been reached.)

I understand and agree.

 

[Peter Leeves]

I haven't had time yet to do those computations in detail, but just looking at the Christoffel symbols, we have $\Gamma_{t \varphi}^{r} \neq 0$, so we would expect a nonzero proper acceleration in the $r$ direction for an object that is rotating about the origin in the $\varphi$ direction. And that means that water in a "non-rotating" bucket in this universe would not have a flat shape! Whereas water in a rotating bucket--one "rotating with the universe"--would have a flat shape!

Good old gut-instinct. I promise not to gloat if it turns out to be correct. It does seem intuitive that water in a rotating bucket in a rotating universe would stay flat. There'd be nothing to cause the water to bulge. On the other hand, if the "hand of god" (or tension in a string) grabbed hold and proper stopped the bucket, the torque I described earlier would be applied causing the water to bulge. The equivalence I've been describing between the two scenarios would be correct. Not that it would negate the coordinate solution discussed earlier. Interesting.

 

[PeterDonis]

The bucket and the water are NOT at rest relative to each other in every scenario we've discussed.

You are mistaken. See below.

The water has to spin up

But you said you understand and agree that we are ignoring the spin-up process and only considering steady state. In steady state the bucket and water are at rest relative to each other.

the reference frame can only be with respect to the non-rotating water

Whether or not the bucket and the water are at rest relative to each other is an invariant; it has nothing to do with any choice of frame and it will be equally true in every frame.

I say this even given the immediately following proviso

I have no idea what you mean by this or why you do not see that your statements are inconsistent with each other, as described above.

 

[PeterDonis]

The equivalence I've been describing between the two scenarios would be correct.

No, it wouldn't. The two spacetime geometries are still different, and that difference will still show up in the two scenarios.

I have not done the detailed computations, but here is what I think they will end up telling us. Note that I am stating everything in terms of invariants and direct observables, with no talk of "frames" at all.

First, we have to be clear about what invariants and direct observables we are talking about. There are actually four:

(1) The proper acceleration of the bucket and the water inside it. This determines the shape of the water's surface (flat for zero proper acceleration, concave for nonzero radial proper acceleration).

(2) The vorticity of the bucket and the water inside it. This determines whether the bucket is rotating (nonzero vorticity) or non-rotating (zero vorticity).

(3) The relative angular velocity of the bucket and the water inside it, with respect to the rest of the matter in the universe. Note that this is not the same as (2) above.

(4) The rotation of the rest of the matter in the universe. This is given by the vorticity of the family of worldlines that describe that matter.

Now let's look at four different scenarios and the invariants for each of them (the first two are known, the last two are what I want to verify by computation):

(NN) Non-rotating bucket in non-rotating universe (flat spacetime). The invariants are:

#1: Zero proper acceleration, flat surface.

#2: Zero vorticity, zero rotation.

#3: Zero relative angular velocity relative to rest of universe.

#4: Zero rotation of rest of universe.

(RN) Rotating bucket in non-rotating universe:

#1: Nonzero proper acceleration, concave surface.

#2: Nonzero vorticity, nonzero rotation.

#3: Nonzero angular velocity relative to rest of universe.

#4: Zero rotation of rest of universe.

(NR) Non-rotating bucket in rotating universe (Godel spacetime):

#1: Nonzero proper acceleration, concave surface.

#2: Zero vorticity, zero rotation.

#3: Nonzero angular velocity relative to rest of universe.

#4: Nonzero rotation of rest of universe.

(RR) Rotating bucket in rotating universe:

#1: Zero proper acceleration, flat surface.

#2: Nonzero vorticity, nonzero rotation.

#3: Zero angular velocity relative to rest of universe.

#4: Nonzero rotation of rest of universe.

Note how the first two invariants, which are the "local" ones, are different in each scenario; they cover all four of the possibilities, given that each of the two observables is a binary choice (zero or nonzero). That means no two of these scenarios are equivalent; each one has a different, unique set of observables.

 

[Peter Leeves]

There's no point repeating myself. If you don't wish to consider my viewpoint, that's absolutely fine. The information I felt was of genuine value and ought to be included in this thread, is now included.

But you said you understand and agree that we are ignoring the spin-up process and only considering steady state. In steady state the bucket and water are at rest relative to each other.

I do understand and agree that we are ignoring the spin-up process and only considering the steady state. At the risk of repeating myself (said I wouldn't but I'll give it one try) I feel the spin-up detail was worth including within this thread for both accuracy and completeness. Particularly for the benefit of others reading this thread at a later date. In short, it genuinely adds some value.

Whether or not the bucket and the water are at rest relative to each other is an invariant; it has nothing to do with any choice of frame and it will be equally true in every frame.

Same comment as above. Costs nothing to note the spin-up and we can all sleep better knowing we've been as accuracte and complete as possible.

I have no idea what you mean by this or why you do not see that your statements are inconsistent with each other, as described above.

My statements are consistent. You wrote a concise and excellent description covering spin up of the water and the reasons it can be ignored. It was this description I referred to as "the proviso". Very well put and definitely worth including in the thread.

 

[Dale]

The bucket and the water are NOT at rest relative to each other in every scenario we've discussed.

The traditional Newton's bucket is at rest relative to the water. I.e. the spin-up or spin-down happened a long time ago, viscous forces have caused all of the relative motion between the water and the bucket and between different parts of the water to dissipate.

"When friction between the water and the sides of the bucket has the two spinning together with no relative motion between them then the water is concave."
https://mathshistory.st-andrews.ac.uk/HistTopics/Newton_bucket/

Edit: never mind, I see that you just discussed that for completeness

 

[PeterDonis]

I do understand and agree that we are ignoring the spin-up process and only considering the steady state.

But you apparently do not agree that in the steady state, the bucket and the water are at rest relative to each other?